We
show in Section 6 (Theorem REF ) that Weyl's submajorization
inequality for the singular
values of a product ( [1]} Theorem 4.2(iii)) continues to hold for
uniform majorization, a notion introduced by Kalton
and Sukochev  [2]}
which is a strengthening of the classical notion of
submajorization in the sense of Hardy, Littlewood and Polya. This permits us to give a simple proof of generalized
Hölder inequalities for general symmetric spaces proved recently by
Bekjan and Ospanov [3]} and Albadawi  [4]}.
